By invoking the correspondence principle of Furstenberg [1]} one deduces that under the previous assumptions on \(a_1,\ldots , a_\ell \) , for every set of integers \(\Lambda \)  and \(\varepsilon >0\)  we have
\(\bar{d}(\Lambda \cap (\Lambda + a_1(n)) \cap \cdots \cap (\Lambda +a_\ell (n)) )\ge (\bar{d}(\Lambda ))^{\ell +1}-\varepsilon \) 
